Christoffel symbol
of a differential quadratic form
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An abbreviated notation for the expression
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The symbols are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind,
, defined by
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where is defined as follows:
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These symbols were introduced by E.B. Christoffel in 1869.
Comments
Let ,
, be a linear connection on a manifold
, where
denotes the space of vector fields on
. Let
be a chart of
. Then on
,
is completely determined by
, where
are coordinates on
. The Christoffel symbols of the connection
are now given by
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It is important to note that the are not the components of a tensor field. In fact if the
denote the Christoffel symbols of
with respect to a second set of coordinates
on
, then
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Let now be the Riemannian connection (cf. Riemannian geometry) defined by a (local) Riemannian metric
. Then the Christoffel symbols of this quadratic differential form are those of the connection
. I.e.,
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so that indeed
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where the are the Christoffel symbols of the second kind of the quadratic differential form as defined above.
References
[a1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 4 |
[a2] | R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. Chapt. 7 |
Christoffel symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_symbol&oldid=41547